If a man gives up parental rights- does he have to pay child support?

It depends on what state you are in. In most states yes, but in some no (unless arrears are owed but that is all that has to be paid). Indiana is one state that there is no support owed if he signs his rights over. The easiest way to know is to call an attourneys office you don't have to pay anything they can just tell you over the phone which way your state rules. Hope that helped.

It will depend on why he is giving them up, First of course normally the judge will not allow you just to give them up. But if a person could get out of child support by giving up their rights plus unless both partners agree you can't just sign them over anyway.

A father can only relinquish his rights if the mother is in agreement and she has a spouse to adopt the child in his place.

If that is the case, when the adoption is final, all current child support will stop. If there is back child support owed, that will still have to be paid unless the mother is willing to forgive that debt.